The oxidation states of transition metal atoms in K₂Cr₂O₇, KMnO₄ and K₂FeO₄, respectively, are x, y and z. The sum of x, y and z is _______.

Two compounds A and B with same molecular formula (C₃H₆O) undergo Grignard’s reaction with methylmagnesium bromide to give products C and D. Products C and D show following chemical tests.
C and D respectively are :

The shape / structure of [XeF₅]^– and XeO₃F₂, respectively, are

Simplified absorption spectra of three complexes ((i), (ii) and (iii)) of Mn⁺ ion are provided below; their $$\lambda $$_max values are marked as A, B and C respectively. The correct match between the complexes and their $$\lambda $$_max values is
(i) [M(NCS)₆]^{(–6 + n)}
(ii) [MF₆]^{(–6 + n)}
(iii) [M(NH₃)₆]ⁿ⁺

The number of subshells associated with n = 4 and m = –2 quantum numbers is

The heat of combustion of ethanol into carbon dioxide and water is – 327 kcal at constant pressure. The heat evolved (in cal) at constant volume and 27^oC (if all gases behave ideally) is (R = 2 cal mol^–1 K^–1) ________.

For the disproportionation reaction
2Cu⁺(aq) ⇌ Cu(s) + Cu²⁺(aq) at 298 K. ln K
(where K is the equilibrium constant) is
___________ × 10^–1.
Given :
($$E_{C{u^{2 + }}/C{u^ + }}^0 = 0.16V$$
$$E_{C{u^ + }/Cu}^0 = 0.52V$$
$${{RT} \over F} = 0.025$$)

The work function of sodium metal is 4.41 $$ \times $$ 10^–19 J. If photons of wavelength 300 nm are incident on the metal, the kinetic energy of the ejected electrons will be (h = 6.63 $$ \times $$ 10^–34 J s; c = 3 $$ \times $$ 10⁸ m/s) ________ × 10^–21 J.

The size of a raw mango shrinks to a much smaller size when kept in a concentrated salt solution. Which one of the following processes can explain this?

The one that is not expected to show isomerism is :

The correct observation in the following reactions is :

The ratio of the mass percentages of ‘C & H’ and ‘C & O’ of a saturated acyclic organic compound ‘X’ are 4 : 1 and 3 : 4 respectively. Then, the moles of oxygen gas required for complete combustion of two moles of organic compound ‘X’ is ________.

Match the type of interaction in column A with the distance dependence of their interaction energy in column B

A	B
(i) ion-ion	(a) $${1 \over r}$$
(ii) dipole-dipole	(b) $${1 \over {{r^2}}}$$
(iii) London dispersion	(c) $${1 \over {{r^3}}}$$
	(d) $${1 \over {{r^6}}}$$

The results given in the below table were obtained during kinetic studies of the following reaction

2A + B $$ \to $$ C + D
X and Y in the given table are respectively :

If you spill a chemical toilet cleaning liquid on your hand, your first aid would be

Three elements X, Y and Z are in the 3^rd period of the periodic table. The oxides of X, Y and Z, respectively, are basic, amphoteric and acidic. The correct order of the atomic numbers of X, Y and Z is :

The molecular geometry of SF₆ is octahedral. What is the geometry of SF₄ (including lone pair(s) of electrons, if any)?

The major product of the following reaction is :

An organic compound ‘A’ (C₉H₁₀O) when treated with conc. HI undergoes cleavage to yield compounds ‘B’ and ‘C’. ‘B’ gives yellow precipitate with AgNO₃ where as ‘C’ tautomerizes to ‘D’. ‘D’ gives positive iodoform test. ‘A’ could be

Arrange the following labelled hydrogens in decreasing order of acidity :

Consider the reaction sequence given below:
Which of the following statements is true?

The major product obtained from E₂ - elimination of 3-bromo-2-fluoropentane is

If the equation cos⁴ $$\theta $$ + sin⁴ $$\theta $$ + $$\lambda $$ = 0 has real solutions for $$\theta $$, then $$\lambda $$ lies in the interval :

Let A = {X = (x, y, z)^T: PX = 0 and

x² + y² + z² = 1} where

$$P = \left[ {\matrix{ 1 & 2 & 1 \cr { - 2} & 3 & { - 4} \cr 1 & 9 & { - 1} \cr } } \right]$$,

then the set A :

If y = $$\sum\limits_{k = 1}^6 {k{{\cos }^{ - 1}}\left\{ {{3 \over 5}\cos kx - {4 \over 5}\sin kx} \right\}} $$,

then $${{dy} \over {dx}}$$ at x = 0 is _______.

For some $$\theta \in \left( {0,{\pi \over 2}} \right)$$, if the eccentricity of the
hyperbola, x²–y²sec²$$\theta $$ = 10 is $$\sqrt 5 $$ times the
eccentricity of the ellipse, x²sec²$$\theta $$ + y² = 5, then the length of the latus rectum of the ellipse, is :

If the sum of first 11 terms of an A.P.,
a₁, a₂, a₃, .... is 0 (a $$ \ne $$ 0), then the sum of the A.P.,
a₁ , a₃ , a₅ ,....., a₂₃ is ka₁ , where k is equal to :

If the variance of the terms in an increasing A.P.,
b₁ , b₂ , b₃ ,....,b₁₁ is 90, then the common difference of this A.P. is_______.

Let f : (–1, $$\infty $$) $$ \to $$ R be defined by f(0) = 1 and
f(x) = $${1 \over x}{\log _e}\left( {1 + x} \right)$$, x $$ \ne $$ 0. Then the function f :

If a curve y = f(x), passing through the point (1, 2), is the solution of the differential equation,
2x²dy= (2xy + y²)dx, then $$f\left( {{1 \over 2}} \right)$$ is equal to :

Let the position vectors of points 'A' and 'B' be
$$\widehat i + \widehat j + \widehat k$$ and $$2\widehat i + \widehat j + 3\widehat k$$, respectively. A point 'P' divides the line segment AB internally in the ratio $$\lambda $$ : 1 ( $$\lambda $$ > 0). If O is the origin and
$$\overrightarrow {OB} .\overrightarrow {OP} - 3{\left| {\overrightarrow {OA} \times \overrightarrow {OP} } \right|^2} = 6$$, then $$\lambda $$ is equal to______.

Let [t] denote the greatest integer less than or equal to t.
Then the value of $$\int\limits_1^2 {\left| {2x - \left[ {3x} \right]} \right|dx} $$ is ______.

Consider a region R = {(x, y) $$ \in $$ R : x² $$ \le $$ y $$ \le $$ 2x}. if a line y = $$\alpha $$ divides the area of region R into two equal parts, then which of the following is true?

The imaginary part of
$${\left( {3 + 2\sqrt { - 54} } \right)^{{1 \over 2}}} - {\left( {3 - 2\sqrt { - 54} } \right)^{{1 \over 2}}}$$ can be :

Let f : R $$ \to $$ R be a function which satisfies
f(x + y) = f(x) + f(y) $$\forall $$ x, y $$ \in $$ R. If f(1) = 2 and
g(n) = $$\sum\limits_{k = 1}^{\left( {n - 1} \right)} {f\left( k \right)} $$, n $$ \in $$ N then the value of n, for which g(n) = 20, is :

Let E^C denote the complement of an event E. Let E₁ , E₂ and E₃ be any pairwise independent events with P(E₁) > 0

and P(E₁ $$ \cap $$ E₂ $$ \cap $$ E₃) = 0.

Then P($$E_2^C \cap E_3^C/{E_1}$$) is equal to :

Let f(x) be a quadratic polynomial such that
f(–1) + f(2) = 0. If one of the roots of f(x) = 0
is 3, then its other root lies in :

$$\mathop {\lim }\limits_{x \to 0} {\left( {\tan \left( {{\pi \over 4} + x} \right)} \right)^{{1 \over x}}}$$ is equal to :

Let n > 2 be an integer. Suppose that there are n Metro stations in a city located along a circular path. Each pair of stations is connected by a straight track only. Further, each pair of nearest stations is connected by blue line, whereas all remaining pairs of stations are connected by red line. If the number of red lines is 99 times the number of blue lines, then the value of n is :

Let a, b, c $$ \in $$ R be all non-zero and satisfy
a³ + b³ + c³ = 2. If the matrix

A = $$\left( {\matrix{ a & b & c \cr b & c & a \cr c & a & b \cr } } \right)$$

satisfies A^TA = I, then a value of abc can be :

The set of all possible values of $$\theta $$ in the interval
(0, $$\pi $$) for which the points (1, 2) and (sin $$\theta $$, cos $$\theta $$) lie
on the same side of the line x + y = 1 is :

The area (in sq. units) of an equilateral triangle inscribed in the parabola y² = 8x, with one of its vertices on the vertex of this parabola, is :

A square shaped hole of side l = $${a \over 2}$$ is carved out at a distance d = $${a \over 2}$$ from the centre ‘O’ of a uniform circular disk of radius a. If the distance of the centre of mass of the remaining portion form O is $$ - {a \over X}$$ , value of X (to the nearest integer) is :

In a Young’s double slit experiment, 16 fringes are observed in a certain segment of the screen when light of a wavelength 700 nm is used. If the wavelength of light is changed to 400 nm, the number of fringes observed in the same segment of the screen would be

A particle of mass m is moving along the x-axis with initial velocity $$u\widehat i$$. It collides elastically with a particle of mass 10 m at rest and then moves with half its initial kinetic energy (see figure). If $$\sin {\theta _1} = \sqrt n \sin {\theta _2}$$ then value of n is ________.

A wire of density 9 $$ \times $$ 10^–3 kg cm^–3 is stretched between two clamps 1 m apart. The resulting strain in the wire is 4.9 $$ \times $$ 10^–4. The lowest frequency of the transverse vibrations in the wire is : (Young’s modulus of wire Y = 9 $$ \times $$ 10¹⁰ Nm^–2), (to the nearest integer), _________

A 10 $$\mu $$F capacitor is fully charged to a potential difference of 50 V. After removing the source voltage it is connected to an uncharged capacitor in parallel. Now the potential difference across them becomes 20 V. The capacitance of the second capacitor is :

If momentum (P), area (A) and time (T) are taken to be the fundamental quantities then the dimensional formula for energy is

The height ‘h’ at which the weight of a body will be the same as that at the same depth ‘h’ from the surface of the earth is (Radius of the earth is R and effect of the rotation of the earth is neglected)

When the temperature of a metal wire is increased from 0^oC to 10^oC, its length increases by 0.02%. The percentage change in its mass density will be closest to :

A wire carrying current I is bent in the shape ABCDEFA as shown, where rectangle ABCDA and ADEFA are perpendicular to each other. If the sides of the rectangles are of lengths a and b, then the magnitude and direction of magnetic moment of the loop ABCDEFA is

Two uniform circular discs are rotating independently in the same direction around their common axis passing through their centres. The moment of inertia and angular velocity of the first disc are 0.1 kg-m² and 10 rad s^–1 respectively while those for the second one are 0.2 kg-m² and 5 rad s^–1 respectively. At some instant they get stuck together and start rotating as a single system about their common axis with some angular speed. The kinetic energy of the combined system is :

An ideal gas in a closed container is slowly heated. As its temperature increases, which of the following statements are true?
(A) the mean free path of the molecules decreases.
(B) the mean collision time between the molecules decreases.
(C) the mean free path remains unchanged.
(D) the mean collision time remains unchanged.

A capillary tube made of glass of radius 0.15 mm is dipped vertically in a beaker filled with methylene iodide (surface tension = 0.05 Nm^–1, density = 667 kg m^–3) which rises to height h in the tube. It is observed that the two tangents drawn from liquid-glass interfaces (from opp. sides of the capillary) make an angle of 60^o with one another. Then h is close to (g = 10 ms^–2)

An ideal cell of emf 10 V is connected in circuit shown in figure. Each resistance is 2 $$\Omega $$. The potential difference (in V) across the capacitor when it is fully charged is ______.

A charge Q is distributed over two concentric conducting thin spherical shells radii r and R (R > r). If the surface charge densities on the two shells are equal, the electric potential at the common centre is :

An inductance coil has a reactance of 100 $$\Omega $$. When an AC signal of frequency 1000 Hz is applied to the coil, the applied voltage leads the current by 45^o. The self-inductance of the coil is

The figure shows a region of length ‘l’ with a uniform magnetic field of 0.3 T in it and a proton entering the region with velocity 4 $$ \times $$ 10⁵ ms^–1 making an angle 60^o with the field. If the proton completes 10 revolution by the time it cross the region shown, ‘l’ is close to
(mass of proton = 1.67 $$ \times $$ 10^–27 kg, charge of the proton = 1.6 $$ \times $$ 10^–19 C)

The displacement time graph of a particle executing S.H.M is given in figure :
(sketch is schematic and not to scale)
Which of the following statements is/are true for this motion?
(A) The force is zero at t = $${{3T} \over 4}$$
(B) The acceleration is maximum at t = T
(C) The speed is maximum at t = $${{T} \over 4}$$
(D) The P.E. is equal to K.E. of the oscillation at t = $${{T} \over 2}$$

In the following digital circuit, what will be the output at ‘Z’, when the
input (A, B) are (1, 0), (0, 0), (1, 1,), (0, 1)

In a hydrogen atom the electron makes a transition from (n + 1)^th level to the n^th level. If n >> 1, the frequency of radiation emitted is proportional to :

In a plane electromagnetic wave, the directions of electric field and magnetic field are represented by $$\widehat k$$ and $$2\widehat i - 2\widehat j$$, respectively. What is the unit vector along direction of propagation of the wave?

A light ray enters a solid glass sphere of refractive index $$\mu $$ = $$\sqrt 3 $$ at an angle of incidence 60^o. The ray is both reflected and refracted at the farther surface of the sphere. The angle (in degrees) between the reflected and refracted rays at this surface is ________.

A particle is moving 5 times as fast as an electron. The ratio of the de-Broglie wavelength of the particle to that of the electron is 1.878 $$ \times $$ 10^–4. The mass of the particle is close to

A small point mass carrying some positive charge on it, is released from the edge of a table. There is a uniform electric field in this region in the horizontal direction. Which of the following options then correctly describe the trajectory of the mass?
(Curves are drawn schematically and are not to scale).